Multicarrier DS/CDMA system using a turbo code with nonuniform repetition coding

ABSTRACT

The present invention relates to a multicarrier direct sequence code division multiple access (DS/CDMA) system using a turbo code with nonuniform repetition coding. In particular, it relates to a multicarrier DS/CDMA system using a turbo code with unequal diversity order in its code symbols instead of using a convolutional code.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a multicarrier direct sequence codedivision multiple access (DS/CDMA) system using a turbo code withnonuniform repetition coding. In particular, it relates to amulticarrier DS/CDMA system using a turbo code with unequal diversityorder in its code symbols instead of using a convolutional code.

2. Description of the Related Art

With the recent development in mobile communication technology, theanalog communication technologies that have been used in prior art, suchas time division multiple access (TDMA) and frequency division multipleaccess (FDMA), are being replaced by code division multiple access(CDMA), a digital communication technology.

CDMA is a method in which a code is assigned to digitized voiceinformation with a broadened frequency band so that each user cantransmit signals according to the code. With this method, a voice signalis first divided into bit units to be encoded, and thereafter insertedinto communication frequency band. As a result, the accommodationcapacity of each channel increases, and the number of users possiblyassigned to the same frequency band is increased by 5–10 times comparedwith the conventional analog-type multiple access technologies. Andthus, it is mainly used for mobile communication technology.

CDMA method is employed in IMT-2000 standard, which is being consideredto be a standard for next-generation mobile communication such asvoice/moving picture data transmission and/or Internet search. However,it has been pointed out that, in CDMA method, the communication qualitycould be degraded due to the excessive interference caused by differentelectric powers assigned to multiple users.

A multicarrier DS/CDMA system using a convolutional code has beenproposed to prevent the above-described quality degradation inestablishing a wideband CDMA system (Y. H. Kim, J. M. Lee, I. H. Song,H. G. Kim and S. C. Kim, “An orthogonal multicarrier DS/CDMA systemusing a convolutional code,” J. of Korean Electro. Soc., vol. 37TC, no.4, pp. 295–303, April 2000; D. N. Rowitch and L. B. Milstein,“Convolutionally coded multicarrier DS CDMA systems in a multipathfading channel-part II: narrowband interference suppression,” IEEETrans. Commun., vol. 47, no. 11, pp. 1729–1736, November 1999). In thissystem, the available bandwidth is divided into a set of disjointequiwidth subchannels, and narrowband DS/CDMA waveforms are transmittedin parallel over the subchannels.

The bit error rate of an orthogonal multicarrier DS/CDMA system using aconvolutional code described above is relatively improved compared withthat of a single-carrier DS/CDMA system in the case that a partialband-interference is being occurred.

In addition to the above-described multicarrier DS/CDMA system using aconvolutional code, a multicarrier DS/CDMA system using a turbo code hasbeen introduced to be a more advanced system (A. H. S. Mohammadi and A.K. Khandani, “Unequal error protection on turbo-encoder output bits,”Electron. Letters, vol. 33, no. 4, pp. 273–274, February 1997; Y. M.Choi and P. J. lee, “Analysis of turbo codes with asymmetricmodulation,” Electron. Letters, vol. 35, no. 1, pp. 35–36, January1999). In this system, different energies are assigned to the two kindsof outputs, information symbols and parity check symbols, of a turboencoder in the additive white Gaussian noise (AWGN) channel to increasethe Euclidean distance, and thus improve the communication quality

That is to say, by using a turbo code instead of a convolutional code,the bit error rate (BER; It indicates the system performance, i.e.communication quality.) is improved such that Eb/No becomes to be 0.7 dBat BER of 10⁻⁵ and rate of ½. Consequently, it provides a communicationquality almost approaching the Shannon limit, and thus, a turbo code isgenerally used for error correction.

However, the prior multicarrier DS/CDMA system using a turbo code has aproblem that a stronger signal energy given to one user causes astronger interference occurring to the other users, and thecommunication quality is degraded thereby.

Besides, in case of inducing more energy to increase the Euclideandistance, the fading effect due to the fading channel cannot be reducedvery much.

SUMMARY OF THE INVENTION

The present invention is proposed to solve the problems of the prior artmentioned above. It is therefore the object of the present invention toprovide a multicarrier DS/CDMA system that improves bit error rate toprovide a better communication quality by using a turbo code instead ofa convolutional code and assigning additional diversity orders to theparity check symbols of a turbo encoder among the two kinds of outputsof a turbo encoder, data bits and parity check symbols.

It is another object of the present invention to increase the number ofpossible users by reducing the fading effect by increasing the diversityof the codes and the channels.

To achieve the object mentioned above, the present invention presents amulticarrier DS/CDMA system using a turbo code with nonuniformrepetition coding comprising: a turbo encoder for encoding input databits; a repeater/symbol mapper for replicating the turbo code symbolsoutputted from the turbo encoder into repetition code symbols infrequency domain and mapping the replicated repetition code symbolsignals appropriately; at least one first interleaver for changingchannels for the turbo code symbols outputted from the repeater/symbolmapper so that the fading effect is properly distributed on the codewordsymbols; at least one first multiplier for multiplying a signaturesequence to the turbo code symbols, which are properly rearranged by atleast one first interleaver, for band-broadening; at least one impulsemodulator for modulating the band-broadened signals from at least onefirst multiplier to an impulse shape; at least one chip wave-shapingfilter for smoothening the waveform of the output signals from at leastone impulse modulator and eliminating the interference between thesymbols; at least one second multiplier for mapping the output signalsfrom at least one chip wave-shaping filter to each frequency band; afirst adder for adding all the output signals from at least one secondmultiplier and transmitting the signals; at least one chip matchedfilter for chip-matched filtering the signals transmitted from the firstadder for each channel; at least one third multiplier for thechip-matched filtered signals for each channel outputted from at leastone chip-matched filter transmitting through; at least one lowpassfilter for inphase modulating and lowpass filtering the signalstransmitted through at least one third multiplier; at least onecorrelator for sampling the lowpass filtered signals outputted from atleast one low-pass filter at each T_(c) and thereafter correlating thesignals with a signature sequence of a user; at least one secondinterleaver for rearranging the sequence of the signals, correlated witha signature sequence of a desired user, outputted from at least onecorrelator back to the original sequence by deinterleaving; a symboldemapper for demapping the separated symbol signals outputted from atleast one second interleaver; and a turbo decoder for decoding thedemapped symbol signals outputted from the symbol demapper.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 a is a systematic diagram of a transmitter of k-th user among Kusers communicating with one another by using a multicarrier directsequence code division multiple access (DS/CDMA) system using a turbocode.

FIG. 1 b is a view illustrating the detail structure of a turbo encoder,described in FIG. 1 a, of which the bit duration is T_(b) and the rateis 1/C.

FIG. 1 c is a view illustrating an example of symbol mapping when usinga turbo code of which M=12 and the rate is ¼ and a repetition vector,which is represented by (M₁, M₂, M₃, M₄), is (2, 5, 3, 2).

FIG. 2 is a systematic diagram of a multicarrier DS/CDMA receiver usinga turbo code.

FIG. 3 a and FIG. 3 b are views illustrating the structure and the stateof a rate ⅓ recursive systematic convolutional (RSC) encoder with agenerator denoted in octal by (1, 5/7, 3/7).

FIG. 4 is a graph showing the union bound and the asymptotic performancewith one bit error rate (BER) term, and the asymptotic performance oftwo BER terms when K=1, M=12, W=100, W=1000 and a rate ⅓ turbo code isused with a repetition vector of (4, 4, 4).

FIG. 5 is a graph showing the bounds with simulation results for threedifferent repetition compositions when K=1, M=12, W=100, W=10000 and arate ⅓ turbo code is used.

FIG. 6 is a graph showing the bit error rate along with E_(b)/η₀ whenK=1, M=12, W=1000 and three turbo codes of different rates are used.

FIG. 7 is a graph showing the bit error rate along with K when M=12,W=1000, E_(b)/η₀=5 dB and a rate ⅓ turbo code is used.

(RSC) Encoder

DETAILED DESCRIPTION OF THE EMBODIMENTS

Hereinafter, referring to appended drawings, the structures and theoperation procedures of the embodiments of the present invention aredescribed in detail.

FIG. 1 a is a systematic diagram of a transmitter of k-th user among Kusers communicating with one another by using a multicarrier DS/CDMAsystem using a turbo code.

As described in FIG. 1 a, a transmitter comprises a turbo encoder (10),a repeater/symbol mapper (20), numbers of first interleavers (30, 31,32), numbers of first multipliers (40, 41, 42), numbers of impulsemodulators (50, 51, 52), numbers of chip wave-shaping filters (60, 61,62), numbers of second multipliers (70, 71, 72) and a first adder (80).

The data bit b_(l) ^((k)) at time 1 for the k-th user with bit durationT_(b) is inputted to a rate 1/C turbo encoder (10), whose output symbolsare denoted by {x_(j,l) ^((k)), j=1, 2, . . . , C}, and encoded therein.

The turbo encoder (10) is described in detail in FIG. 1 b.

FIG. 1 b is a view illustrating the detail structure of a turbo encoder,described in FIG. 1 a, of which the bit duration is T_(b) and the rateis 1/C.

As described in FIG. 1 b, a turbo encoder (10) comprises an interleaver(1), a first recursive systematic convolutional (RSC) encoder (2) and asecond recursive systematic convolutional (RSC) encoder (3).

The rates of the first RSC encoder (2) and the second RSC encoder (3)are 1/C₁ and 1/C₂, respectively.

For the outputs of the encoder, x_(1,l) ^((k)) denotes thesystematically transmitted binary data symbol, {x_(j,l)^((k))}_(p1)={x_(2,l) ^((k)), . . . , x_(C) ₁ _(,l) ^((k))} denote theparity check symbols encoded by the first RSC encoder (2), and {x_(j,l)^((k))}_(p2)={x_(C) ₁ _(+1,l) ^((k)), . . . , x_(C,l) ^((k))} denote theparity check symbols from the second RSC encoder (3) in which the inputsignal b_(l) ^((k)), which is properly rearranged by the interleaver(1), is encoded.

The symbol x_(j,l) ^((k)) is repetition coded at a rate 1/M_(j) andmapped into the subchannels by the repeater/symbol mapper (20) toproperly separate, in frequency, the repetition code symbols of eachturbo code symbol.

Here, an example of symbol mapping by using a symbol mapper is describedin FIG. 1 c.

FIG. 1 c is a view illustrating an example of symbol mapping when usinga turbo code of which M=12 and the rate is ¼ and a repetition vector,which is represented by (M₁, M₂, M₃, M₄), is (2, 5, 3, 2).

As described in FIG. 1 c, it illustrates an embodiment of mapping 4turbo code symbols to 12 sub-channels when a repetition vector (M₁, M₂,M₃, M₄) is (2, 5, 3, 2).

Assume that {{tilde over (x)}_(m,l) ^((k)), m=1, 2, . . . , M} are thesignals outputted from the symbol mapper.

Here, {tilde over (x)}_(m,l) ^((k)) is the symbol transmitted throughthe m-th subchannel at time l . More specifically, {tilde over(x)}_(m,l) ^((k)) is equal to x_(j,l) ^((k)) if m belongs to the indexset A_(j) whose elements are the indexes of the subchannels wherex_(j,l) ^((k)) is transmitted.

Numbers of first interleavers (30, 31, 32) described in FIG. 1 arearranges the signals {{tilde over (x)}_(m,l) ^((k)), m=1, 2, . . . ,M} supplied by the repeater/symbol mapper (20) by channel-interleavingin order that the deep fading effect is properly distributed on thecodeword symbols.

Then, numbers of first multipliers (40, 41, 42) multiply signaturesequences to the turbo code symbols, which are properly rearranged bynumbers of first interleavers (30, 31, 32) and outputs band-broadenedsignals.

Numbers of impulse modulators (50, 51, 52) modulate the band-broadenedsignals from numbers of first multipliers (40, 41, 42) into impulseshape and outputs the modulated signals.

In-phase modulators and orthogonal-phase modulators are combined to beused as impulse modulators in a prior multicarrier DS/CDMA system usinga convolutional code, however, in-phase modulators are only used asnumbers of impulse modulators (50, 51, 52) in the present invention forsimplifying the system components.

Numbers of chip wave-shaping filters (60, 61, 62) smoothen the waveformof the impulse-type moderated signals outputted from numbers of impulsemodulators (50, 51, 52), eliminates the interference between symbols andoutputs them.

Numbers of second multipliers (70, 71, 72) are mapping the outputsignals from numbers of chip wave-shaping filters (60, 61, 62) to eachfrequency band.

A first adder (80) adds all the output signals from numbers of secondmultipliers (70, 71, 72) and transmits them to a receiver.

Here, the transmission signal of a turbo code symbol of k-th user isshown in Equation 1.

$\begin{matrix}{{{s^{(k)}(t)} = {\sqrt{2E_{c}}{\sum\limits_{m = 1}^{M}\;{\sum\limits_{n = {- \infty}}^{\infty}\;{{\overset{\sim}{x}}_{m,{\lbrack{n/N}\rbrack}}^{(k)}c_{n}^{(k)}{p\left( {t - {nT}_{c}} \right)}{\cos\left( {{\omega_{m}t} + \varphi_{m}^{(k)}} \right)}}}}}},} & \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$where,

E_(c): energy-per-chip,

P(t): impulse response of chip wave-shaping filter,

{c_(n) ^((k))}: random binary signature sequence with period N,

1/T_(c): chip rate,

ω_(m): angular frequency of m-th subcarrier, and

{φ_(m) ^((k))}: independent identically distributed (iid) randomvariables uniformly distributed over [0, 2π).

Here, each subchannel is assumed to be a slowly varying frequencynonselective Rayleigh fading with impulse response function representedby Equation 2.h _(m) ^((k))(t)=α_(m) ^((k)) e ^(jθ) ^(m) ^((k)) δ(t), m=1, 2, . . . ,M,  [Equation 2]where,

{α_(m) ^((k))}: fading amplitudes, and

{θ_(m) ^((k))}: random phases.

As shown in the above equation, the fading amplitudes and phases arecorrelated in time and frequency.

However, the correlation occurred in the fading amplitudes can bereduced by using proper channel interleavers (30, 31, 32) and a symbolmapper.

That is to say, in case that the fading amplitudes are independentidentically distributed (iid) Rayleigh random variables with a unitsecond moment and the phases are iid random variables uniformlydistributed over [0, 2π), the receiving signal is given by Equation 3.

$\begin{matrix}\begin{matrix}{{r(t)} = {\sqrt{2E_{c}}{\sum\limits_{k = 1}^{K}{\sum\limits_{m = 1}^{M}{\sum\limits_{n = {- \infty}}^{\infty}\;{\alpha_{m}^{(k)}{\overset{\sim}{x}}_{m{\lbrack{n/N}\rbrack}}^{(k)} c_{n}^{(k)}{p\left( {t - {\quad{{nT}_{c} -}}} \right.}}}}}}} \\{\left. \quad\tau_{k} \right){\cos\left( {{{\omega_{m} t} + \left. \quad\phi_{m}^{(k)} \right) + {n_{W}(t)}},} \right.}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack\end{matrix}$where,

{τ_(k)} (propagation delay): independent identically distributed (iid)random variables uniformly distributed over [0, T_(b)),φ_(m) ^((k)): φ_(m) ^((k))+θ_(m) ^((k))−ω_(m)τ_(k), and

n_(W)(t): additive white Gaussian noise (AWGN) having power spectraldensity (psd) of η₀/2.

FIG. 2 is a systematic diagram of a receiver of k-th user among K userscommunicating with one another by using a multicarrier DS/CDMA systemusing a turbo code.

As described in FIG. 2, a receiver comprises numbers of chip matchedfilters (90, 91, 92), numbers of third multipliers (100, 101, 102),numbers of lowpass filters (110, 111, 112), numbers of correlators (120,121, 122), numbers of second interleavers (140, 141, 142), a symboldemapper (150), and a turbo decoder (160).

Numbers of chip matched filters (90, 91, 92) perform a chip matchedfiltering on the received signals, transmitted from the transmitter ofthe other users, for each channel.

Numbers of lowpass filters (110, 111, 112) modulate the chip matchedfiltered signal for each channel supplied through numbers of thirdmultipliers (100, 101, 102) to be in-phased and perform a lowpassfiltering on them.

Here, the outputs of the lowpass filters are sampled every T_(c) andthen correlated with the signature sequence of a desired user by numbersof correlators (120, 121, 122).

Numbers of second interleavers (140, 141, 142) rearrange the sequence ofthe signals outputted from numbers of correlators (120, 121, 122) backto the original sequence by channel deinterleaving, a symbol demapper(150) is demapping the separated symbol signals, and a turbo decoder(160) decodes the signals.

The chip wave-shaping filter (60), P(f), of the above describedtransmitter satisfies the Nyquist criterion and is of unit energy. Here,it is defined that g(t)=F⁻¹{G(f)}, where, G(f)=|P(f)|² and F⁻¹ denotesthe inverse Fourier transform.

Decoding procedures in the turbo decoder (160) are now described indetail.

The inputs for an iterative turbo decoder are modeled assuming that thefirst user (k=1) is a desired user and that perfect carrier, code andbit synchronization is obtained. Then, the correlator output of the q-thsubchannel at time 1 can be written as Equation 4.

$\begin{matrix}\begin{matrix}{Z_{q,l}^{(1)} = {\sum\limits_{n^{\prime} = 0}^{N - 1}\;{c_{n^{\prime}}^{(1)}{y_{q}\left( {{lT}_{b} + {n^{\prime}T_{c}} + \tau_{1}} \right)}}}} \\{{= {{N\sqrt{E_{c}}\alpha_{q}^{(1)}{\overset{\sim}{x}}_{q,l}^{1}} + U_{q,l} + W_{q,l}}},}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$where,

y_(q)(t): the lowpass filter output on the q-th subchannel after beingconverted to the baseband,

U_(q,1): the multiple access interference (MAI) output of the q-thcorrelator output at time 1,

W_(q,1): the additive white Gaussian noise (AWGN) output of the q-thcorrelator output at time 1.

In has been shown in the prior art that Z_(q,l) ⁽¹⁾, conditioned onα_(q) ⁽¹⁾ and {tilde over (x)}_(q,l) ¹, is approximately Gaussian, wherethe approximation is valid for a large number of users and is perfectfor K=1.

The means of the random variables, U_(q,1) and W_(q,1), are 0 (zero),and their variances are given by Equation 5.

$\begin{matrix}\begin{matrix}{{{{Var}\left( U_{q,l} \right)} = {\frac{{N\left( {K - 1} \right)}E_{c}}{2T_{c}}{\int_{- \infty}^{\infty}{{{G(f)}}^{2}\ {\mathbb{d}f}}}}},{and}} \\{{{Var}\left( W_{q,l} \right)} = {\frac{N\;\eta_{0}}{2}.}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

Using the equation (Eqn. 5), decoder inputs can be modeled. The decoderinput model can be written by Equation 6.z _(q,l)=α_(q,l) {tilde over (x)} _(q,l) +I _(q,l),   [Equation 6]where,

z_(q,l), I_(q,l): the correlator output and its total noise componentrespectively, normalized by N√{square root over (E_(c))}, on the q-thsubchannel at time 1 after channel deinterleaving.

Here, user indexes on the superscript of symbols are dropped fornotational convenience. The MAI plus AWGN term, I_(q,l), isapproximately a zero mean Gaussian random variable, and its variance isgiven by Equation 7.

$\begin{matrix}{{\sigma_{q}^{2} = {{\frac{K - 1}{2{NT}_{c}}{\int_{- \infty}^{\infty}{{{G(f)}}^{2}\ {\mathbb{d}f}}}} + \frac{\eta_{0}M}{2E_{b}}}},} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$where,

E_(b)(=NME_(C)): energy-per-bit.

Before turbo decoding is performed, the normalized correlator outputsare partitioned into C subsets, {z_(l) ^(j), j=1, 2, . . . , C}, by thesymbol demapper.

Here, z_(l) ^(j)={z_(q,l), qεA_(J)} is the subset composed of thecorrelator outputs for the subchannels where the turbo code symbol,x_(j,1), is transmitted.

The symbol mapping shown in FIG. 3 is denoted by the followingequations:z_(l) ¹={z_(3,l), z_(10,l)},z_(l) ²={z_(1,l), z_(4,l), z_(6,l), z_(8,l), z_(11,l)},z_(l) ³={z_(2,l), z_(7,l), z_(12,l)}, andz_(l) ⁴={z_(5,l), z_(9,l)}.

The C subsets are again divided into three groups: z_(l) ¹ correspondingto the systematic data symbol, {z_(l) ^(j)}_(p) ₁ ={z_(l) ², . . . ,z_(l) ^(C) ¹ } corresponding to the parity check symbols of the firstRSC decoder, and {z_(l) ^(J)}_(p) ₂ ={z_(l) ^(C) ¹ ⁺¹, . . . , z_(l)^(C)} corresponding to the parity check symbols of the second RSCdecoder.

When a rate ¼ turbo code described in FIG. 3 consists of a rate ½ firstRSC code (2) and a rate ⅓ second RSC code (3), it follows that {z_(l)^(J)}_(p) ₁ ={z_(l) ²} and {z_(l) ^(j)}_(p) ₂ ={z_(l) ³, z_(l) ⁴}.

These sets are inputted to a turbo decoder (160) along with theinformation on the fading amplitudes and noise variances.

Turbo decoding metric on multicarrier DS/CDMA system model using a turbocode with nonuniform repetition coding will be described hereinafter.

The turbo decoder (160) uses an iterative, suboptimal, soft-decodingrule where each constituent RSC code is separately decoded by itsdecoder. However, the constituent decoders share bit-likelihoodinformation through an iterative process.

In addition, maximum a posteriori (MAP) algorithm is used for both thefirst and the second RSC decoders with some modification on the branchtransition metric to incorporate the appropriate channel statistics forthe system model.

For a fully interleaved channel with known fading amplitudes α={α_(1,1),. . . , α_(M,1), . . . , α_(1,W), . . . , α_(M,W)} with the block sizeof the input data sequence, W, the branch transition metric can bewritten by Equation 8.γ₁((z _(l) ¹ , {z _(l) ^(J)}_(p) ₁ ),S _(l−1) ,S _(l)|α)=q(b _(l) =i|S_(l−1) ,S _(l))p(z _(l) ¹ ,{z _(l) ^(j)}_(p) ₁ |b _(l) =i,S _(l−1) ,S_(l)α)Pr{S _(l) |S _(l−1)},   [Equation 8]where,

S₁: the encoder state at time 1.

Here, the state can take values between [0, 2^(v)−1] for an RSC encoderhaving v memory elements.

In other words, the value of q(b_(l)=i|S_(l−1), S_(l)) is 1 if atransition from states S_(l−1) to S_(l) occurs, and is 0 otherwise. Theprobability, Pr{S_(l)|S_(l−1)}, is given by Equation 9.

$\begin{matrix}{{\Pr\left\{ S_{l} \middle| S_{l - 1} \right\}} = \left\{ \begin{matrix}{{\Pr\left\{ {b_{l} = 1} \right\}},} & {{{{when}\mspace{14mu}{q\left( {{b_{l} = \left. 1 \middle| S_{l - 1} \right.},S_{l}} \right)}} = 1},} \\{{\Pr\left\{ {b_{l} = 0} \right\}},} & {{{{when}\mspace{14mu}{q\left( {{b_{l} = \left. 0 \middle| S_{l - 1} \right.},S_{l}} \right)}} = 1},} \\{0,} & {{otherwise}.}\end{matrix} \right.} & \left\lbrack {{Equation}\mspace{14mu} 9} \right\rbrack\end{matrix}$

Here, a priori probabilities, Pr{b_(l)=i} are estimated by the otherdecoder through the iterative decoding process.

The conditional probability, p(●|●), appeared in Equation 8, is given byEquation 10.

$\begin{matrix}\begin{matrix}{{p\left( {z_{l}^{1},{\left. \left\{ z_{l}^{j} \right\}_{p_{1}} \middle| b_{l} \right. = i},S_{l - 1},S_{l},\alpha} \right)} = {p\left( {\left. z_{l}^{1} \middle| b_{l} \right. =} \right.}} \\{{\left. {i,S_{l - 1},S_{l},\alpha} \right){p\left( {{\left. \left\{ z_{l}^{j} \right\}_{p_{1}} \middle| b_{l} \right. = i},S_{l - 1},S_{l},\alpha} \right)}} =} \\{\prod\limits_{q \in A_{1}}{\frac{1}{\sqrt{2\pi\;\sigma_{q}^{2}}}\exp\left\{ {- \frac{\left( {z_{q,l} - {\alpha_{q,l}{x_{1,l}(i)}}} \right)^{2}}{2\sigma_{q}^{2}}} \right\} \times}} \\{{\prod\limits_{j = 2}^{C_{1}\;}{\prod\limits_{q \in A_{j}}{\frac{1}{\sqrt{2\pi\;\sigma_{q}^{2}}}\exp\left\{ {- \frac{\left( {z_{q,l} - {\alpha_{q,l}{x_{j,l}\left( {i,S_{l - 1},S_{l}} \right)}}} \right)^{2}}{2\sigma_{q}^{2}}} \right\}}}} =} \\{B_{l}\exp{\left\{ {{\left( {\sum\limits_{q \in A_{1}}\;\frac{\alpha_{q,l}z_{q,l}}{\sigma_{q}^{2}}} \right) x_{1,l}( i)} + {\sum\limits_{j = 2}^{C_{1}}\;{\left( {\sum\limits_{q \in A_{j}}\;\frac{\alpha_{q,l}z_{q,l}}{\sigma_{q}^{2}}} \right){x_{j,l}\left( {i, S_{l - 1}, S_{l}} \right)}}}} \right\}.}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 10} \right\rbrack\end{matrix}$

Here, x_(1,l)(i)=±1 according to b₁=i; x_(j,l)(i, S_(l−1), S_(l))=±1according to the value of the j-th code symbol generated when the statetransits from S_(l−1) to S₁ with input, b_(l)=i; and B₁ is a constantwhich has no influence on the decoding process.

In a similar way, the transition metric for the second RSC decoder canbe obtained with the interleaved version of z_(l) ¹ and the other setsof parity check outputs, {z_(l) ^(J)}_(p) ₂ .

With the transition metrics described above, the log-likelihood ratio(LLR) of the l-th data bit, b₁, is computed at each decoder by Equation11.

$\begin{matrix}\begin{matrix}{{L_{k}\left( {\hat{b}}_{l} \right)} = {\log\frac{\Pr\left\{ {b_{l} = \left. 1 \middle| {observation} \right.} \right\}}{\Pr\left\{ {b_{l} = \left. 0 \middle| {observation} \right.} \right\}}}} \\{\mspace{14mu}{{= {\log\frac{\sum\limits_{\; S_{l}}^{\;}\;{\sum\limits_{S_{l - 1}}^{\;}\;{{\gamma_{1}\left( {\left( {z_{l}^{1},\left\{ z_{l}^{j} \right\}_{p_{k}}} \right),S_{l - 1},\left. S_{l} \middle| \alpha \right.} \right)}{ɛ_{l - 1}\left( S_{l - 1} \right)}{\beta_{l}\left( S_{l} \right)}}}}{\sum\limits_{\; S_{l}}^{\;}\;{\sum\limits_{S_{l - 1}}^{\;}\;{{\gamma_{0}\left( {\left( {z_{l}^{1},\left\{ z_{l}^{j} \right\}_{p_{k}}} \right),S_{l - 1},\left. S_{l} \middle| \alpha \right.} \right)}{ɛ_{l - 1}\left( S_{l - 1} \right)}{\beta_{l}\left( S_{l} \right)}}}}}},}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 11} \right\rbrack\end{matrix}$where,

L_(k)(●): LLR computed by the decoder k(=1, 2),

ε_(l)(●): forward recursion, and

β_(l)(●): backward recursion.

Here, the forward recursion is generally denoted by α_(l)(●), however,it is replaced by ε₁(●) in Equation 11 to avoid confusion with thefading amplitudes.

With the branch transition metric and LLR obtained by Equation 11, codesymbols are decoded through an iterative process.

The turbo code is also denoted by a (C(W+v),W) block code. Here, W isthe size of the input data sequence and Cv tail bits are appended toterminate the first RSC encoder state to the all-zero state. The effectof tail bits can be neglected when W is much larger than v, the memorysize of constituent code.

It can be assumed that the all-zero codewordx={x_(1,1),Λ,x_(C,1),Λ,x_(1,W),Λ,x_(C,W)} is transmitted since the turbocode is a linear code.

Then, the union bound on the probability of word error under maximumlikelihood (ML) decoding can be written as Equation 12.

$\begin{matrix}{{P_{W} \leq {\sum\limits_{d = 1}^{CW}\;{{A(d)}{P_{2}(d)}}}},} & \left\lbrack {{Equation}\mspace{14mu} 12} \right\rbrack\end{matrix}$where,

A(d): number of codewords with Hamming weight d, and

P₂(d): pairwise error probability that the ML decoder will prefer aparticular codeword {circumflex over (x)} of total Hamming weight d tothe all-zero codeword x.

Since it is difficult to obtain A(d) for a specific interleaver, anaverage upper bound constructed with a random interleaver is proposed inthe prior art (D. Divsalar, S. Dolinar, and F. Pollara, “Transferfunction bounds on the performance of turbo codes,” TDA Progress Report42–122, Jet Propul. Lab., Pasadena, USA, pp. 44–55, August 1995).

From the results of the prior art mentioned above, the average weightdistribution is given by Equation 13.

$\begin{matrix}{{{\overset{\_}{A}(d)} = {\sum\limits_{i = 1}^{W}\;{\begin{pmatrix}W \\i\end{pmatrix}{p\left( d \middle| i \right)}}}},} & \left\lbrack {{Equation}\mspace{14mu} 13} \right\rbrack\end{matrix}$where,

p(d|i): probability that an input data sequence with Hamming weight iproduces a codewoed with Hamming weight d.

Thus, the average union bound is given by Equation 14.

$\begin{matrix}{{\overset{\_}{P}}_{W} \leq {\sum\limits_{i = 1}^{W}\;{\begin{pmatrix}W \\i\end{pmatrix}{\sum\limits_{d = 1}^{CW}{p\;\left( d \middle| i \right){{P_{2}(d)}.}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack\end{matrix}$

Similarly, the average union bound on the probability of bit error isgiven by Equation 15.

$\begin{matrix}{{\overset{\_}{P}}_{b} \leq {\sum\limits_{i = 1}^{W}{\frac{i}{W}{\sum\limits_{d = 1}^{CW}{p\;\left( d \middle| i \right){{P_{2}(d)}.}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 15} \right\rbrack\end{matrix}$

In the system of the present invention, the C distinct turbo codesymbols for a given input symbol may go through channels with differentnoise levels and diversity orders.

Thus, it has to be taken into consideration how a codeword Hammingweight is constructed from the different kinds of turbo code symbols.

For this purpose, a turbo codeword {circumflex over (x)} is divided intoC code fragments. Here, the j-th fragment is given by {circumflex over(x)}^(J)={{circumflex over (x)}_(J,1), {circumflex over(x)}_(J,2),Λ,{circumflex over (x)}_(J,W)}. If d_(j) denotes the Hammingweight of {circumflex over (x)}^(J), Equation 15 can be rewritten byEquation 16.

$\begin{matrix}{{\overset{\_}{P}}_{b} \leq {\underset{d_{1} = 1}{\overset{W}{\quad\sum}}\frac{d_{1}}{W}\begin{pmatrix}W \\d_{1}\end{pmatrix}{\sum\limits_{\underset{{d{({= {\sum\limits_{j = 1}^{C}\; d_{j}}})}} = 1}{({d_{2},d_{3},\Lambda,d_{C}})}}{p\;\left( {d_{2},d_{3},\Lambda,\left. d_{C} \middle| d_{1} \right.} \right){{P_{2}\left( {d_{1},d_{2},\Lambda,d_{C}} \right)}.}}}}} & \left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack\end{matrix}$

Here, p(d₂,d₃,Λ,d_(C)|d₁) is the probability that an input data sequenceof weight d₁ produces a codeword of total weight d=Σ_(J=1) ^(C)d_(j)with the code fragment weight d_(j) of {circumflex over (x)}^(j), andP₂(d₁,d₂,Λ,d_(C)) is the pairwise error probability that the ML decoderchooses a particular codeword {circumflex over (x)}^(J) with fragmentweights (d₁,d₂,Λ,d_(C)) rather than the all-zero codeword.

In case that a random interleaver (i.e. random sequence) is used in thesystem described in FIG. 2, an input data sequence of weight d₁ israndomly permuted to some other sequence of weight d₁ at the input ofthe second RSC encoder.

That is to say, the permutation does not depend on the pattern of inputdata sequence on where the nonzero bits are located. And thus, the codefragments generated by the second RSC encoder with the permuted datasequence is different from those generated by the first RSC encoder withthe original input data sequence as in the prior art by D. Divsalar, etal.

The probability p(d₂,Λ,d_(C)|d₁) is therefore decomposed into Equation17.p(d ₂ ,Λ,d _(C) |d ₁)=p ₁(d ₂ ,Λ,d _(C) ₁ |d ₁)p ₂(d _(C) ₁ ₊₁ ,Λ,d _(C)|d _(l)).   [Equation 17]

Here, p₁(d_(s) _(i) ,Λ,d_(e) _(i) |d₁) is the probability that the RSCencoder i produces the codeword fragments {{circumflex over(x)}^(J)}_(J=S) ₁ ^(e) ¹ of weights (d_(s) _(i) ,Λ,d_(e) _(i) ) withinput data sequence of weight d₁. And, s_(i) and e_(i) denote thestarting index and the ending index, respectively, of the codewordfragments {circumflex over (x)}^(J) produced by the RSC encoder i. Theprobability p_(i)(d_(s) _(i) ,Λ,d_(e) _(i) |d₁) is given by Equation 18.

$\begin{matrix}{{{p_{i}\;\left( {d_{s_{i}},\Lambda,\left. d_{e_{i}} \middle| d_{1} \right.} \right)} = \frac{t_{i}\left( {W,d_{1},d_{s_{i}},\Lambda,d_{e_{i}}} \right)}{\begin{pmatrix}W \\d_{1}\end{pmatrix}}},{i = 1},2,} & \left\lbrack {{Equation}\mspace{14mu} 18} \right\rbrack\end{matrix}$where,

$\begin{pmatrix}W \\d_{1}\end{pmatrix}\text{:}$number of all input data sequences of weight d₁, and

t_(i)(W,d₁,d_(s) _(i) ,Λ,d_(e) _(i) ): number of codewords whose codefragments produced by the RSC encoder i have the weights (d_(s) _(i),Λ,d_(e) _(i) ).

Here, t_(l)(●) is obtained by using the transfer function methodproposed in the prior art by D. Divsalar, et al with some modificationsuch that an RSC code transfer function enumerates all the fragmentweights.

FIG. 3 a and FIG. 3 b are views illustrating the structure and the stateof a rate ⅓ RSC encoder with a generator denoted in octal by (1, 5/7,3/7).

In FIG. 3 b, dummy variables are used to specify the path (L), the inputweight (D₁), and the weight of code fragments (D₂, D₃) corresponding tothe parity check symbols.

With a similar method to that of the prior art by D. Divsalar, et al,the state transition matrix of the code is obtained as Equation 19.

$\begin{matrix}{{A\left( {L,D_{1},D_{2},D_{3}} \right)} = {\quad{\begin{bmatrix}L & {L\; D_{1}D_{2}} & 0 & 0 \\0 & 0 & {L\; D_{1}D_{3}} & {L\; D_{2}D_{3}} \\{L\; D_{1}D_{2}D_{3}} & {L\; D_{3}} & 0 & 0 \\0 & 0 & {L\; D_{2}} & {L\; D_{1}}\end{bmatrix}.}}} & \left\lbrack {{Equation}\mspace{14mu} 19} \right\rbrack\end{matrix}$

Using Equation 19, the transfer function is derived as Equation 20.

$\begin{matrix}\begin{matrix}{{T\left( {L,D_{1},D_{2},D_{3}} \right)} = \left\lfloor \left( {I - {A\left( {L,D_{1},D_{2},D_{3}} \right)}} \right)^{- 1} \right.} \\\left. {A\left( {L,1,D_{2},D_{3}} \right)}^{v} \right\rfloor_{({0,0})} \\{= {\sum\limits_{l \geq 0}{\sum\limits_{d_{1} \geq 0}{\sum\limits_{d_{2} \geq 0}\sum\limits_{d_{3} \geq 0}}}}} \\{{t\left( {l,d_{1},d_{2},d_{3}} \right)}L^{l}D_{1}^{d_{1}}D_{2}^{d_{2}}{D_{3}^{d_{3}}.}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 20} \right\rbrack\end{matrix}$

Here, (0,0) denotes the (0,0) element of the matrix and the termA(L,1,D₂,D₃) represents the contribution of tail bits in which neitherpath nor input data weight is accumulated. Using the transfer functionobtained in this way, t(W,d₁,d₂,d₃) can be obtained by the recursion asdescribed in the prior art by D. Divsalar, et al.

Besides, the pairwise error probability P₂(d₁,d₂,Λ,d_(C)) can beobtained when the channel side information such as fading amplitudes, α,and noise variance at each subchannel, {σ₉ ²,q=1, 2,Λ,M}, is available.

That is, the conditional probability of incorrectly decoding theall-zero codeword x into a particular codeword {circumflex over (x)}with code fragment weights (d₁,d₂,Λ,d_(C)), conditioned on α, is givenby Equation 21.

$\begin{matrix}{{{P\left( {x,\left. \hat{x} \middle| \alpha \right.} \right)} = {Q\left( \sqrt{\sum\limits_{j = 1}^{C}\;{\sum\limits_{l \in \xi_{j}}{\sum\limits_{q \in A_{j}}{\alpha_{q,l}^{2}/\sigma_{q}^{2}}}}} \right)}},} & \left\lbrack {{Equation}\mspace{14mu} 21} \right\rbrack\end{matrix}$where,

${{Q(x)} = {\frac{1}{\sqrt{2\;\pi}}{\int_{x}^{\infty}{{\mathbb{e}}^{- \frac{u^{2}}{2}}\ {\mathbb{d}u}}}}},$and

ξ_(J): index set of 1 such that x_(J,) _(l) ≠{circumflex over(x)}_(J,l).

Here,d_(j) is the cardinality of ξ_(J), since a codeword fragment{circumflex over (x)}^(J) differs in d_(j) places from the all-zerocodeword fragment x^(J).

Thus, the average pairwise error probability, averaged over α, is givenby Equation 22.

$\begin{matrix}{{{P_{2}\left( {d_{1},d_{2},\Lambda,d_{C}} \right)} = {\int_{0}^{\infty}{\Lambda{\int_{0}^{\infty}{{Q\left( \sqrt{2\;{\mu\left( {x,\left. \hat{x} \middle| \alpha \right.} \right)}} \right)}{\prod\limits_{j = 1}^{C}\;{\prod\limits_{l \in \xi_{j}}^{\;}\;{\prod\limits_{q \in A_{j}}^{\;}\;{{p_{\alpha}\left( \alpha_{q,l} \right)}d\;\alpha_{q,l}}}}}}}}}},} & \left\lbrack {{Equation}\mspace{14mu} 22} \right\rbrack\end{matrix}$where,μ(x,{circumflex over (x)}|α)=Σ_(J=1) ^(C)Σ_(lεξ) _(J) Σ_(qεA) _(J)α_(q,l) ²/2σ_(q) ², andp _(α)(y)=2ye ^(−y) ² , for y≧0.

Since it is difficult to obtain the exact P₂(d₁,d₂,Λ,d_(C)) for allpossible values of (d₁,d₂,Λ,d_(C)), a bound on the pairwise errorprobability is obtained instead by using the method proposed by Slimaneet al (S. B. Slimane and T. Le-Ngoc, “Tight bounds on the errorprobability of coded modulation schemes in Rayleigh fading channels,”IEEE Trans. Vehic. Technol., vol. 44, no. 1, pp. 121–130, February1995).

In this method, it is defined that α_(q,l)=α_(q,l)√{square root over(1+½σ_(q) ²)}, and thus Equation 23 can be derived.

$\begin{matrix}\begin{matrix}{{{P_{\alpha}\left( \alpha_{q,l} \right)}{{d\;\alpha_{q,l}}}} = {2\alpha_{q,l}{\mathbb{e}}^{- \alpha_{q,l}^{2}}{{d\;\alpha_{q,l}}}}} \\{= {\frac{{\mathbb{e}}^{a_{q,l}^{2}{({{2\sigma_{q}^{2}} + 1})}}}{1 + {{1/2}\sigma_{q}^{2}}}2a_{q,l}{\mathbb{e}}^{- a_{q,l}^{2}}{{{d\; a_{q,l}}}.}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 23} \right\rbrack\end{matrix}$

By using Equation 23, Equation 22 can be rewritten as Equation 24.

$\begin{matrix}{{{P_{2}\left( {d_{1},d_{2},\Lambda,d_{C}} \right)} = {\prod\limits_{j = 1}^{C}\;{\prod\limits_{q \in A_{j}}^{\;}\;{\left( \frac{1}{1 + {{1/2}\sigma_{q}^{2}}} \right)^{d_{j}}{\int_{0}^{\infty}{\Lambda{\int_{0}^{\infty}{{Q\left( \sqrt{2{\mu\left( {x,{\hat{x}\left. a \right)}} \right.}} \right)}{\mathbb{e}}^{\mu({x,{\hat{x}{a)}}}}{\prod\limits_{j = 1}^{C}\;{\prod\limits_{l = 1}^{d_{j}}\;{\prod\limits_{q \in A_{j}}^{\;}{{p_{\alpha}\left( a_{q,l} \right)}{\mathbb{d}a_{q,l}}}}}}}}}}}}}},} & \left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack\end{matrix}$where,μ(x,{circumflex over (x)}|α)=Σ_(J=1) ^(C)Σ_(lεξ) _(J) Σ_(qεA) _(J)α_(q,l) ²/(2σ₉ ²+1), and

p_(α)(a_(q, l)) = 2a_(q, l)𝕖^(−a_(q, l)²),for α_(q,l)≧0.

Besides, defining the minimum and the maximum values among the {σ_(q) ²,q=1, 2,Λ,M} to be σ_(min) ² and σ_(max) ², it follows that w_(min)²v≦μ(x, {circumflex over (x)}|α)≦w_(max) ²v, where

${w_{\min} = \sqrt{\frac{1}{{2\sigma_{\max}^{2}} + 1}}},{w_{\max} = \sqrt{\frac{1}{{2\sigma_{\min}^{2}} + 1}}},$and v=Σ_(J=1) ^(C)Σ_(lεξ) _(J) Σ_(qεA) _(J) α_(q,l) ².

More specifically, the random variable v is the sum of d_(t)=Σ_(J=1)^(C)d_(J)M_(J) Rayleigh random variables with unit second moment. Thatis, v is a chi-square random variable with 2d_(t) degrees of freedom.

Thus, Equation 24 is bounded by Equation 25.

$\begin{matrix}{{{P_{2}\left( {d_{1},d_{2},\Lambda,d_{C}} \right)} \leq {{K\left( {d_{t},w_{\min}} \right)}{\prod\limits_{j = 1}^{C}\;{\prod\limits_{q \in A_{j}}^{\;}\;\left( \frac{1}{1 + {{1/2}\sigma_{q}^{2}}} \right)^{d_{j}}}}}},} & \left\lbrack {{Equation}\mspace{11mu} 25} \right\rbrack\end{matrix}$where,

${K\left( {d_{t},w} \right)} = {{\int_{0}^{\infty}{{Q\left( {w\sqrt{2v}} \right)}{\mathbb{e}}^{w^{2}v}{p_{v}(v)}{\mathbb{d}v}}} = {\frac{1}{2^{2d_{t}}}{\sum\limits_{k = 1}^{d_{t}}{\begin{pmatrix}{{2d_{t}} - k - 1} \\{d_{t} - 1}\end{pmatrix}{\frac{2^{k}}{\left( {1 + w} \right)^{k}}.}}}}}$

In this way, the average bit error probability is bounded as Equation26.

$\begin{matrix}{{\overset{\_}{P}}_{b} \leq {\sum\limits_{d_{1} = 1}^{W}{\frac{d_{1}}{W}\begin{pmatrix}W \\d_{1}\end{pmatrix}{\sum\limits_{\underset{{d{({= {\sum\limits_{j = 1}^{C}d_{j}}})}} = 1}{({d_{2},d_{3},\Lambda,d_{C}})}}^{\;}{\begin{matrix}{p\left( {{d_{2,}d_{3,}\Lambda},{d_{C}{d_{1}}}} \right)}\end{matrix}\begin{matrix}{K\left( {d_{t},w_{\min}} \right){\prod\limits_{j = 1}^{C}\;{\prod\limits_{q \in A_{j}}^{\;}\;{\left( \frac{1}{1 + {{1/2}\sigma_{q}^{2}}} \right)^{d_{j}}.}}}}\end{matrix}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 26} \right\rbrack\end{matrix}$

Since it is difficult to calculate K(d_(t),w_(min)) in Equation 26 forlarge values of d_(t) due to the numerical problem and calculation time,Equation 26 is further upper bounded as Equation 27 by using the factthat K(d,w) is a monotonically decreasing function of d_(t).

$\begin{matrix}{{{\overset{\_}{P}}_{b} \leq {{K\left( {d_{t,\min},w_{\min}} \right)}{\sum\limits_{d_{1} = 1}^{W}{\frac{d_{1}}{W}\begin{pmatrix}W \\d_{1}\end{pmatrix}{\sum\limits_{\underset{{d{({= {\sum\limits_{j = 1}^{C}d_{j}}})}} = 1}{({d_{2},d_{3},\Lambda,d_{C}})}}^{\;}{\begin{matrix}{p\left( {{d_{2,}d_{3,}\Lambda},{d_{C}{d_{1}}}} \right)}\end{matrix}{\prod\limits_{j = 1}^{C}\;{\prod\limits_{q \in A_{j}}^{\;}\;\left( \frac{1}{1 + {{1/2}\sigma_{q}^{2}}} \right)^{d_{j}}}}}}}}}},} & \left\lbrack {{Equation}\mspace{14mu} 27} \right\rbrack\end{matrix}$where

$d_{t,\min} \equiv {\min\limits_{({d_{1},d_{2},\Lambda,d_{C}})}{\sum\limits_{j = 1}^{C}{d_{j}{M_{j}:}}}}$total minimum distance of the codewords resulting after repetitioncoding is applied to the turbo code symbols.

Here, if the noise variance is the same for all the subchannels suchthat σ_(q) ²=σ², Equation 27 can be simplified as Equation 28.

$\begin{matrix}{{\overset{\_}{P}}_{b} \leq {{K\left( {d_{t,\min},w_{\min}} \right)}{\sum\limits_{d_{1} = 1}^{W}{\frac{d_{1}}{W}\begin{pmatrix}W \\d_{1}\end{pmatrix}{\sum\limits_{\underset{{d{({= {\sum\limits_{j = 1}^{C}d_{j}}})}} = 1}{({d_{2},d_{3},\Lambda,d_{C}})}}^{\;}{\begin{matrix}{p\left( {{d_{2,}d_{3,}\Lambda},{d_{C}{d_{1}}}} \right)}\end{matrix}{\left( \frac{1}{1 + {{1/2}\sigma^{2}}} \right)^{\sum\limits_{j = 1}^{C}{d_{j}M_{j}}}.}}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 28} \right\rbrack\end{matrix}$

Even though average union bounds are known to accurately estimate theperformance of a turbo code in the moderate to high signal-to-noiseratio (SNR) region, called the error floor region, it is difficult toobtain the number of codewords for all possible code fragment weightsfor large values of W and C. Thus, in the present invention, theperformance of a turbo code is estimated by an asymptotic performancewhich shows the behavior of the BER in the moderate to high SNR regionas the random interleaver gets large.

There are two important factors of turbo codes: the inetrleaving(permutation) of input data sequences and the recursiveness of theconstituent codes. The interleaver matches the low weight output of oneencoder with the high weight output of the other, thereby reduces theprobability that the net codeword has low weights.

On the other hand, the recursiveness of the constituent codes eliminatesthe codewords due to input data sequences of weight 1 which remain inthe same pattern after interleaving. Thus, the minimum weight of inputdata sequences is 2 for turbo coding, and it is less likely for turbocodewords to have low weight as either the weight of the input datasequences or the size of the interleaver increases.

That is to say, the probability that an input data sequence of weight 2which generates the lowest output weight at the first RSC encoder ispermuted to the pattern which generates the lowest output weight at thesecond RSC encoder is approximately 2/W, while the probability for aninput data sequence of weight 3 is approximately 6/W² for randominterleaving.

Thus, the minimum weight of the codewords due to input data sequences ofweight 2, defined as the effective free distance, dominates theperformance in the moderate SNR region as the size of the randominterleaver gets large.

On the other hand, the BER term due to the codewords of total minimumdistance d_(t,min), dominates the performance in the high SNR region,since the BER terms in Equation 26 decrease fast as the SNR increases atthe d_(t) power of the inverse SNR, and thus the BER terms due tod_(t)(>d_(t,min)) are negligible compared to that due to d_(t,min).

Based on these observations, the asymptotic performance considering thetwo BER terms due to the effective free distance and total minimumdistance in Equation 26 is given by Equation 29.

$\begin{matrix}{{{\overset{\_}{P}}_{b,{asy}} \equiv {{B_{ef}{K\left( {d_{t,{ef}},w_{\min}} \right)}{\prod\limits_{j = 1}^{C}\;{\prod\limits_{q \in A_{j}}\;\left( \frac{1}{1 + {{1/2}\sigma_{q}^{2}}} \right)^{d_{jef}}}}} + {B_{\min}{K\left( {d_{t,\min},w_{\min}} \right)}{\prod\limits_{j = 1}^{C}\;{\prod\limits_{q \in A_{j}}\;\left( \frac{1}{1 + {{1/2}\sigma_{q}^{2}}} \right)^{d_{j,\min}}}}}}},} & \left\lbrack {{Equation}\mspace{14mu} 29} \right\rbrack\end{matrix}$where,

$B_{ef} = {\frac{2}{W}\begin{pmatrix}W \\2\end{pmatrix}{p\left( {d_{2,{ef}},\Lambda,{d_{C,{ef}}\left. 2 \right)},{B_{\min} = {\frac{d_{1,\min}}{W}\begin{pmatrix}{W\mspace{25mu}} \\d_{1,\min}\end{pmatrix}{p\left( {d_{2,\min},\Lambda,{d_{C,\min}\left. d_{1,\min} \right)},} \right.}}}} \right.}}$

d_(j,ef): j-th code fragment weight contributing to the effective freedistance d_(ef)=Σ_(J=1) ^(C)d_(J,ef),

d_(t,ef)=2M₁+Σ_(J=2) ^(C)d_(J,ef)M_(J): total effective free distanceafter the repetition coding is applied to the turbo code symbols, and

d_(j,min): j-th code fragment weight contributing to the total minimumdistance d_(t,min).

FIG. 4 compares the union bound, the asymptotic performance with one BERterm due to the effective free distance (Asymp 1), and the asymptoticperformance with two BER terms due to the effective free distance andtotal minimum distance (Asymp 2) when K=1, M=12, W=100, W=10000 and arate ⅓ turbo code is used with a repetition vector of (4, 4, 4).

As described in FIG. 4, while Asymp 1 deviates from the union bound athigh SNR when W=100, Asymp 2 quite well estimates the union bound.

Thus, it will be enough to find the minimum distance and the effectivefree distance of a turbo code to estimate the performance in the errorfloor region.

In addition, the performance of the proposed system is evaluated in thepresent invention for various repetition vectors (M₁, M₂,Λ, M_(C)).Either the average union bound or the asymptotic performance is used toestimate the performance in the error floor region.

Various simulations were also carried out in the present invention forthe low SNR region where neither the bound nor the asymptoticperformance can be applied. Random interleavers are used for thepermutation of input data sequences, and the number of decodingiterations is 15 for all simulation results.

It is assumed that G(f) is a raised-cosine filter with rolloff factor0.5, and the total processing gain due to spreading and coding is fixedsuch that PG_(t)=NM=512 for all simulation results.

Here, the number M of subcarriers is first chosen, and N is thendetermined by the largest integer less than PG_(t)/M. The generatingpolynomial of a turbo code is denoted by (1,g₂/h₂,Λ,g_(C)/h_(C)), whereg_(j) and h_(j) are the forward and the feedback polynomials in octalfor the j-th parity check symbols.

First, the performance is investigated when K=1 (i.e. single user case).

FIG. 5 compares the results from simulations with the bounds for threerepetition combinations when W=100 and W=1000 and a rate ⅓ turbo codewith generator (1, 5/7, 5/7) is used.

The turbo code used for performance investigation is the one with thebest effective free distance among the codes of the same memory.

In FIG. 5, it is observed that the bounds fall slightly below thesimulation results, which, in part, results from the fact that thebounds are based on ML decoding and the simulation results are obtainedwith the suboptimal iterative decoding algorithm.

Such a behavior was also observed in the prior art (S. Benedetto and G.Montorsi, “Design of parallel concatenated convolutional codes,” IEEETrans. Commun., vol. 44, pp. 591–600, may 1996), where the convergenceof the simulated performance toward the analytical bounds was shown foran AWGN channel.

Another observation is that the system with a (2, 5, 5) repetitionvector performs better in the error floor region as W increases. Thiseffect arises because the effective free distance dominates theperformance as W increases.

The turbo code used for FIG. 5 has an effective free distance d_(ef)=10,whereas the code fragment weights are d_(1,ef)=2,d_(2,ef)=4, andd_(3,ef)=4.

After repetition coding is applied, the total effective free distancebecomes d_(t,ef)=36,d_(t,ef)=40, and d_(t,ef)=44 for the (6, 3, 3), (4,4, 4), and (2, 5, 5) repetition vectors, respectively. Thus, the BER atwhich the error floor begins can be lowered by giving more diversity tothe parity check symbols and thereby increasing the total effective freedistance.

In addition, FIG. 5 shows the simulation results of the performance ofthe three repetition methods in the low SNR region when W=1000.

It is observed that the repetition vector processing better performancein the error floor region shows somewhat worse performance for very lowSNRs. The reason for such behavior may be explained by the followingconjecture.

The log-likelihood ratio (LLR) in Equation 11 can be partitioned asEquation 30.L _(k)({circumflex over (b)} _(l))=L _(c,k)({circumflex over (b)}_(l))+L _(e,k)({circumflex over (b)} _(l))+L _(p,k)({circumflex over(b)} _(l)),where

${L_{c,k}\left( {\hat{b}}_{l} \right)} = {2{\sum\limits_{q \in A_{1}}\;{\frac{\alpha_{q,l}z_{q,l}}{\sigma_{q}^{2}}\text{:}}}}$constituent due to the channel measurement of the systematic data,

${L_{e,k}\left( {\hat{b}}_{l} \right)} = {\log\frac{\left. {\sum\limits_{S_{l}}{\sum\limits_{S_{l - 1}}{{\gamma_{1}\left( {\left\{ z_{l}^{j} \right\}_{p_{k}},S_{l - 1},S_{l}} \right.}\alpha}}} \right){ɛ_{l - 1}\left( S_{l - 1} \right)}{\beta_{l}\left( S_{l} \right)}}{\left. {\sum\limits_{S_{l}}{\sum\limits_{S_{l - 1}}{{\gamma_{0}\left( {\left\{ z_{l}^{j} \right\}_{p_{k}},S_{l - 1},S_{l}} \right.}\alpha}}} \right){ɛ_{l - 1}\left( S_{l - 1} \right)}{\beta_{l}\left( S_{l} \right)}}\text{:}}$extrinsic information obtained from the decoding process, and

L_(p,k)({circumflex over (b)}_(l)): a priori probability which is givenby the extrinsic information from the other decoder in the iterativedecoding process.

It is expected that the decision of the data bit becomes more reliableby the use of both L_(e,k)({circumflex over (b)}_(l)) andL_(p,k)({circumflex over (b)}_(l)) through the iterative decodingprocess. However, at low SNR values, one decoder outputs unreliableerroneous extrinsic information, which adds erroneous information to theother decoder, and thus it is hard for the decoders to converge on adecision of the transmitted data.

Thus, if a higher diversity order is given to the systematic data, alower diversity order is given to the parity check symbols, and theeffect of erroneous information exchange will be reduced in the low SNRregion.

However, as the decoder begins to perform better as the SNR increases,the extrinsic information from one decoder enhances the ability of theother decoder to out put more reliable LLR, and vice versa.

Thus, there is a tradeoff between the performance in the low SNR regionand that in the high SNR region according to how much diversity is givento the data symbols and how much is given to the parity check symbols.

FIG. 6 is a graph showing the bit error rate along with E_(b)/η₀ whenK=1, M=12, W=1000 and three turbo codes of different rates are used, andTable 1 shows the parameters and total effective distances for someturbo codes of rate 1/C constructed from two RSC codes of rate 1/C₁ and1/C₂.

TABLE 1 1/C (1/C₁, 1/C₂) (1, g₂/h₂, . . . , g_(C)/h_(C)) (M₁, . . . ,M_(C)) (d_(1,ef), . . . , d_(C,ef)) d_(t,ef) 1/3 (1/2, 1/2) (1, 5/7,5/7) (2, 5, 5) (2, 4, 4) 44 1/4 (1/2, 1/3) (1, 5/7, 5/7, 3/7) (2, 5, 3,2) (2, 4, 4, 2) 40 1/5 (1/3, 1/3) (1, 5/7, 3/7, 5/7, 3/7) (2, 3, 2, 3,2) (2, 4, 2, 4, 2) 36

FIG. 6 shows the effect of the different kinds of parity check symbolsby fixing the repetition number of the data symbols and changing thecode rate and the repetition number of the parity check symbols. Inaddition, the parameters and the repetition vectors of the codes usedfor obtaining FIG. 6 are provided in Table 1 along with thecorresponding total effective free distance.

While the rate ⅓ turbo code with the (2, 5, 5) repetition vectorperforms best in the error floor region, it shows the worst performancein the low SNR region. Note that the code fragment generated by thegenerator 3/7 has a smaller weight contribution to the effective freedistance than that of the generator 5/7. Thus, as new code fragmentsgenerated by the generator 3/7 are added, the performance in the errorfloor region gets worse, while that in the low SNR region gets better.Therefore, the desired code rate and repetition method should beproperly selected according to the operating BER values.

It is also investigated in the present invention the performance of thesystem when the multiple access interference (MAI) exists.

FIG. 7 is a graph showing the bit error rate along with K when M=12,W=1000, E_(b)/η₀=5 dB and a rate ⅓ turbo code is used.

In the figure, it is observed that there is a crossover between the tworepetition methods, which was also observed in the single user case.

That is, at the BER values below the crossover, the effective freedistance dominates the performance characteristics so that the systemwith the (2, 5, 5) repetition vector performs better than that with the(4, 4, 4) repetition vector. On the other hand, the results are reversedat the BER values above the crossover, since increasing the diversityorder to the parity check symbols leads to some divergence of theiterative decoding in the low SNR region.

Conclusively, by increasing the total effective free distance with aproper repetition method, the system capacity can be increased at thelow BER region near or below the error floor region.

For example, at a BER of 10⁻⁵, the system can admit about 40 users morewith the (2, 5, 5) repetition vector than with the (4, 4, 4) repetitionvector for a rate ⅓ turbo code.

As mentioned thereinbefore, a multicarrier DS/CDMA system in accordancewith the present invention increases the diversity order to the paritycheck symbols when encoding/decoding the data bits in a turboencoder/decoder so that it can provide a better communication qualityunder the mobile communication environment when being applied to a basestation and a mobile terminal.

In addition, it can increase the number of possible users under themobile communication environment by reducing the fading effect byincreasing the diversity of the codes and the channels.

Since those having ordinary knowledge and skill in the art of thepresent invention will recognize additional modifications andapplications within the scope thereof, the present invention is notlimited to the embodiments and drawings described above.

1. A multicarrier direct sequence code division multiple access(DS/CDMA) system using a turbo code with nonuniform repetition codingcomprising: a turbo encoder for encoding input data bits; arepeater/symbol mapper for replicating the turbo code symbols outputtedfrom said turbo encoder into repetition code symbols in frequency domainand mapping said replicated repetition code symbol signalsappropriately; at least one first interleaver for changing channels forthe turbo code symbols outputted from said repeater/symbol mapper sothat the fading effect is properly distributed on the codeword symbols;at least one first multiplier for multiplying a signature sequence tothe turbo code symbols, which are properly rearranged by said at leastone first interleaver, for band-broadening; at least one impulsemodulator for modulating the band-broadened signals from said at leastone first multiplier to an impulse shape; at least one chip wave-shapingfilter for smoothening the waveform of the output signals from said atleast one impulse modulator and eliminating the interference between thesymbols; at least one second multiplier for mapping the output signalsfrom said at least one chip wave-shaping filter to each frequency band;a first adder for adding all the output signals from said at least onesecond multiplier and transmitting the signals; and a reception systemfor receiving the transmitted signals from said first adder.
 2. Amulticarrier direct sequence code division multiple access (DS/CDMA)system using a turbo code with nonuniform repetition coding as claimedin claim 1, wherein said reception system comprises: at least one chipmatched filter for chip-matched filtering the signals transmitted fromsaid first adder for each channel; at least one third multiplier for thechip-matched filtered signals for each channel outputted from said atleast one chip-matched filter transmitting through; at least one lowpassfilter for inphase modulating and lowpass filtering the signalstransmitted through said at least one third multiplier; at least onecorrelator for sampling the lowpass filtered signals outputted from saidat least one low-pass filter at each T_(c) and thereafter correlatingthe signal with a signature sequence of a user; at least one secondinterleaver for rearranging the sequence of the signals, correlated witha signature sequence of a desired user, outputted from said at least onecorrelator back to the original sequence by deinterleaving; a symboldemapper for demapping the separated symbol signals outputted from saidat least one second interleaver; and a turbo decoder for decoding thedemapped symbol signals outputted from said symbol demapper.
 3. Amulticarrier direct sequence code division multiple access (DS/CDMA)system using a turbo code with nonuniform repetition coding as claimedin claim 1, wherein said turbo encoder comprises: an interleaver; afirst recursive systematic convolutional (RSC) encoder for encodinginput turbo code symbols into parity check symbols; and a secondrecursive systematic convolutional (RSC) encoder for encoding the turbocode symbols properly rearranged by said interleaver into parity checksymbols, and thereby characterize by performing a repetition coding withgiving an additional diversity order to the turbo code symbols.
 4. Amulticarrier direct sequence code division multiple access (DS/CDMA)system using a turbo code with nonuniform repetition coding as claimedin claim 1, wherein said impulse modulators are constructed fromin-phase modulators.
 5. A multicarrier direct sequence code divisionmultiple access (DS/CDMA) system using a turbo code with nonuniformrepetition coding as claimed in claim 2, characterized in that saidturbo decoder decodes the turbo code symbols by using a turbo decodingmetric of an iterative process based on a branch transition metric and alog-likelihood ratio (LLR) using the following equations of Equations 31and 32: $\begin{matrix}{{\left. {{{{p\left( {z_{l}^{1},\left\{ z_{l}^{j} \right\}_{p_{1}}} \right.}b_{l}} = i},S_{l - 1},S_{l},\alpha} \right) = {{p\left( {{{z_{l}^{1}\left. {{b_{l} = i},S_{l - 1},S_{l},\alpha} \right){p\left( \left\{ z_{l}^{j} \right\}_{p_{1}} \right.}b_{l}} = i},S_{l - 1},S_{l},\alpha} \right)} = {{\prod\limits_{q \in A_{1}}{\frac{1}{\sqrt{2{\pi\sigma}_{q}^{2}}}\exp\left\{ {- \frac{\left( {z_{q,l} - {\alpha_{q,l}{x_{1,l}(i)}}} \right)^{2}}{2\sigma_{q}^{2}}} \right\} \times {\prod\limits_{j = 2}^{C_{1}}\;{\prod\limits_{q \in A_{j}}{\frac{1}{\sqrt{2{\pi\sigma}_{q}^{2}}}\exp\left\{ {- \frac{\left( {z_{q,l} - {\alpha_{q,l}{x_{j,l}\left( {i,S_{l - 1},S_{l}} \right)}}} \right)^{2}}{2\sigma_{q}^{2}}} \right\}}}}}} = {B_{l}\exp\left\{ {{\left( {\sum\limits_{q \in A_{1}}\;\frac{\alpha_{q,l}z_{q,l}}{\sigma_{q}^{2}}} \right){x_{1,l}(i)}} + {\sum\limits_{j = 2}^{C_{1}}{\left( {\sum\limits_{q \in A_{j}}\;\frac{\alpha_{q,l}z_{q,l}}{\sigma_{q}^{2}}} \right){x_{j,l}\left( {i,S_{l - 1},S_{l}} \right)}}}} \right\}}}}},} & \left\lbrack {{Equation}\mspace{14mu} 31} \right\rbrack\end{matrix}$  where, x_(1,l)(i)=±1 according to b₁=i;x_(J,l)(i,S_(l−1),S_(l))=±1 according to the value of the j-th codesymbol generated when the state transits from S_(l−1) to S₁ with input,b_(l)=i; and B₁: a constant which has no influence on the decodingprocess, $\begin{matrix}{{{L_{k}\left( {\hat{b}}_{l} \right)} = {{\log\frac{\left. {\Pr\left\{ {b_{l} = 1} \right.{observation}} \right\}}{\left. {\Pr\left\{ {b_{l} = 0} \right.{observation}} \right\}}} = {\log\frac{\left. {\sum\limits_{S_{l}}{\sum\limits_{S_{l - 1}}{{\gamma_{1}\left( {\left( {z_{l}^{1},\left\{ z_{l}^{j} \right\}_{p_{k}}} \right),S_{l - 1},S_{l}} \right.}\alpha}}} \right){ɛ_{l - 1}\left( S_{l - 1} \right)}{\beta_{l}\left( S_{l} \right)}}{\left. {\sum\limits_{S_{l}}{\sum\limits_{S_{l - 1}}{{\gamma_{0}\left( {\left( {z_{l}^{1},\left\{ z_{l}^{j} \right\}_{p_{k}}} \right),S_{l - 1},S_{l}} \right.}\alpha}}} \right){ɛ_{l - 1}\left( S_{l - 1} \right)}{\beta_{l}\left( S_{l} \right)}}}}},} & \left\lbrack {{Equation}\mspace{14mu} 32} \right\rbrack\end{matrix}$  where, L_(k)(●): LLR computed by the decoder k(=1, 2),ε₁(●): forward recursion, and β₁(●): backward recursion.